Ferric nitrate is the result of the reaction of nitric acid and iron. It provides a way to get iron ions into the wood without the excitement associated with reacting iron and nitric acid. The heating process causes the nitrate half of the ferric nitrate to snag onto a hydrogen ion to become nitric acid. Nitric acid boils (evaporates) at approximately 180 F so heating the stock causes the nitric acid to evaporate, leaving the iron ions behind. The iron ions go through a couple of reactions that eventually result in ferric oxide, iron oxide, with the typical orangy-brown/brown/blackish-brown rust color. The color varies somewhat due to the formation of hydrates with the water that is part of the wood.
If the color is not as dark as you expect I would first try reheating the wood. It might be that the initial heating did not fully drive off the nitrate ions as nitric acid. Another possibility is that the concentration of the iron nitrate solution was not sufficient to carry enough iron into the wood. Reapplication of the solution will add additional iron ions but you will have to reheat the wood to drive off the nitrate ions, like before. My experience leads me to believe that the more iron you get into the wood the more intense the coloration. Iron oxide is not soluble in water so it doesn't wash out.
I have never used tannic acid. I know what it is. It is a poly-phenol with carboxylic acid groups on the perimeter of the molecule. But I don't know if or how it works, although Mitch Yates says it does and he knows far more than I do.
I have found that when I use iron acetate solution for stain (aka vinegar and iron) that sometimes the stock comes out blacker than I want without the red/orange undertones. Treating the stock with hydrogen peroxide pushes the reactions towards the orangy-brown color as does heating and leaving the stock dry for a few days. The blackish color slowly progresses towards the orangy-brown.
That's all I know. Maybe it is some help to you.
Best Regards,
John Cholin
